NEW YORK — A month after a bitter fight over Viacom’s leadership was settled, the company may find itself pressed to reunite with its corporate sibling.

National Amusements — which is owned by the billionaire Sumner M. Redstone and is the controlling shareholder in Viacom and CBS — plans to send letters to both companies, instructing them to consider a merger, people briefed on the matter said Wednesday.

The request would prompt CBS and Viacom to establish independent board committees to explore the idea, said the people, who spoke on the condition of anonymity. As of Wednesday afternoon, neither company had received such a letter.

The instructions could be the first step toward joining the two media companies, which split in 2006 during a period of questions over leadership and the growth of both businesses.

Their fortunes have diverged since then: CBS, buoyed by strong ratings at its television network, has enjoyed a rising stock price and support from investors. By contrast, Viacom has struggled as properties like Comedy Central and MTV have suffered from slumping ratings.

This year, Shari Redstone, Redstone’s daughter, clashed with Philippe P. Dauman, then Viacom’s chief executive, over the direction of the company in a bitter and unusually public battle. At one point, after changes were made on the boards of National Amusements and Viacom, there were legal battles in three states in the fight for control. Then in August, Dauman agreed to step down as chairman and chief executive.

Even now, Viacom is experiencing turmoil and uncertainty about its future. Its interim chief executive, Tom Dooley, said he would step down in November, leaving the company to find a permanent leader quickly.

Having the implicit backing of National Amusements, the movie chain theater started by Sumner Redstone’s father, could add impetus to the idea of combining CBS and Viacom again. National Amusements controls about 80 percent of the voting shares in Viacom and CBS.

Some analysts have said that a merger makes sense, creating a bigger company with greater negotiating clout over cable and satellite providers.

They have also contended that CBS’ longtime and well-respected chief executive, Leslie Moonves, could be better positioned to fix the issues at Viacom as the top executive overseeing both companies.

But Moonves has often expressed doubt about a corporate reunion. A spokesman for CBS said in an emailed statement Wednesday, “As we’ve said before, the CBS Corporation will always act in the best interest of all of its shareholders.’’

Representatives for Viacom and National Amusements declined to comment. Shares in Viacom were up 3 percent, and CBS rose 4 percent Wednesday, after Reuters reported National Amusements’ intentions.